Question

what is ( mwidehat{po} )?
( 128^{circ} )
( 173^{circ} )
( 192^{circ} )
( 256^{circ} )

Explanation:

Step1: Use the formula for the measure of an angle formed by two secants

The formula is \( \angle N=\frac{1}{2}(m\widehat{PO}-m\widehat{PM})\). Let \(m\widehat{PO}=x\) and \(m\widehat{PM} = 360 - x\) (since the sum of the measures of a major arc and a minor arc in a circle is \(360^{\circ}\)). We know that \(\angle N = 45^{\circ}\).
Substituting into the formula: \(45=\frac{1}{2}(x-(360 - x))\).

Step2: Simplify the equation

First, expand the right - hand side: \(45=\frac{1}{2}(x - 360+x)\).
Then \(45=\frac{1}{2}(2x-360)\).
Multiply both sides by 2: \(90 = 2x-360\).
Add 360 to both sides: \(2x=90 + 360=450\) (This is wrong. Let's start from the correct formula. The formula for the angle formed by a secant and a tangent (if it was, but here it's two secants, the correct formula is \(\angle N=\frac{1}{2}(m\widehat{PO}-m\widehat{MO})\). Given \(m\widehat{MO} = 83^{\circ}\) and \(\angle N=45^{\circ}\))
Using the formula \(\angle N=\frac{1}{2}(m\widehat{PO}-m\widehat{MO})\), substitute \(\angle N = 45^{\circ}\) and \(m\widehat{MO}=83^{\circ}\).
We get \(45=\frac{1}{2}(m\widehat{PO}-83)\).
Multiply both sides by 2: \(90=m\widehat{PO}-83\).
Add 83 to both sides: \(m\widehat{PO}=90 + 83=173^{\circ}\).

Answer:

\(173^{\circ}\)